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The topological nature of nodal lines in a three-band system can be describedby non-Abelian topological charges called quaternion numbers. Due to the gaugefreedom of the eigenstates, the sign of the non-Abelian topological charges canbe flipped by performing the gauge transformation, i.e., choosing a differentbasis of eigenstates. However, the sign flipping has not been explicitly shownin realistic systems such as phononic and photonic topological insulators.Here, we elaborate the sign freedom by visualizing the numerically calculatedtopological charges in phononic and photonic topological insulators. In doingso, we employ a common reference point method for multiple nodal lines toconfirm that the sign flipping does not cause any issue in building thequaternion group.

Sign freedom of non-Abelian topological charges in phononic and photonic topological insulators